2.2 Brillouin zones

Brillouin zones are the primitive cells of reciprocal space, and they set the stage for nearly everything in solid state physics: electronic band structures, phonon dispersion, Fermi surfaces, and diffraction conditions. If you understand how Brillouin zones are constructed and why their boundaries matter, the rest of the course clicks into place much more naturally.

Brillouin zones in reciprocal space

A Brillouin zone is the Wigner-Seitz primitive cell constructed in the reciprocal lattice rather than in real space. The reciprocal lattice itself is the Fourier transform of the real-space Bravais lattice, so every point in reciprocal space corresponds to a spatial frequency of the crystal's periodicity.

Because the reciprocal lattice is periodic, any wave vector $\mathbf{k}$ can be mapped back into the first Brillouin zone by adding or subtracting reciprocal lattice vectors. This means the first zone contains all physically distinct $\mathbf{k}$-points. Everything outside it is just a copy.

First Brillouin zone boundaries

The first Brillouin zone is defined as the set of all points in reciprocal space that are closer to the origin than to any other reciprocal lattice point. Its boundaries are the perpendicular bisector planes (Bragg planes) of the vectors connecting the origin to its nearest reciprocal lattice neighbors.

A wave with wave vector $\mathbf{k}$ lying exactly on a zone boundary satisfies the Bragg condition. At that point the wave is coherently scattered by the periodic potential, which is precisely why energy gaps open at zone boundaries. So the boundary isn't just a geometric curiosity; it's where diffraction happens and band gaps appear.

Brillouin zone construction

Wigner-Seitz cell in reciprocal space

The construction procedure is identical to building a Wigner-Seitz cell in real space, just carried out in reciprocal space:

1. Start at the origin of the reciprocal lattice.
2. Draw vectors $\mathbf{G}$ from the origin to all neighboring reciprocal lattice points.
3. For each $\mathbf{G}$, construct the plane that perpendicularly bisects it (i.e., the set of points satisfying $\mathbf{k} \cdot \hat{\mathbf{G}} = |\mathbf{G}|/2$).
4. The smallest enclosed volume around the origin is the first Brillouin zone.
5. Higher-order zones (2nd, 3rd, etc.) are the successive shells formed by the next set of bisector planes.

Because the construction uses the full symmetry of the reciprocal lattice, the resulting Brillouin zone automatically inherits that symmetry (the point group of the lattice).

Brillouin zones vs lattice planes

Relationship of Brillouin zones to Bragg planes

Every Brillouin zone boundary is a Bragg plane. The Bragg condition in real space is:

$2d\sin\theta = n\lambda$

Translating this into reciprocal space: a reciprocal lattice vector $\mathbf{G}$ is perpendicular to a set of lattice planes with spacing $d = 2\pi / |\mathbf{G}|$. The perpendicular bisector of $\mathbf{G}$ is exactly the locus of wave vectors that satisfy the diffraction condition for those planes. So hitting a zone boundary means the wave is Bragg-reflected. This is why the nearly-free electron model predicts band gaps at zone boundaries: the incoming and reflected waves mix, creating standing waves with different energies.

Brillouin zones of common lattices

Cubic lattice Brillouin zones

• Simple cubic (SC): The reciprocal lattice is also simple cubic with spacing $2\pi/a$. The first Brillouin zone is a cube of side $2\pi/a$ centered at the origin. Key symmetry points: $\Gamma$ (center), $X$ (face center), $M$ (edge center), $R$ (corner).
• BCC: The reciprocal lattice is FCC. The first Brillouin zone is a rhombic dodecahedron (12 faces). Key points include $\Gamma$, $H$, $N$, and $P$.
• FCC: The reciprocal lattice is BCC. The first Brillouin zone is a truncated octahedron (14 faces: 8 hexagons and 6 squares). Key points include $\Gamma$, $X$, $L$, $W$, $U$, and $K$.

The FCC Brillouin zone is the one you'll encounter most often because many technologically important materials (Si, GaAs, Cu, Al) have FCC-based structures.

Hexagonal lattice Brillouin zones

For a hexagonal lattice, the first Brillouin zone is a hexagonal prism. It has six rectangular side faces and two hexagonal top/bottom faces. High symmetry points are labeled $\Gamma$ (zone center), $M$ (midpoint of a rectangular face edge), $K$ (corner of the hexagonal face), $A$ (center of the top hexagonal face), $L$, and $H$.

The $K$ point is especially important in 2D hexagonal systems like graphene, where the Dirac cones sit at the $K$ and $K'$ points.

Wave propagation in periodic structures

Dispersion relation in periodic lattices

The dispersion relation $\omega(\mathbf{k})$ (or equivalently $E(\mathbf{k})$) tells you how frequency or energy depends on wave vector. In a periodic potential, this relation is itself periodic in reciprocal space with the periodicity of the reciprocal lattice:

$E(\mathbf{k} + \mathbf{G}) = E(\mathbf{k})$

This periodicity is why all unique information fits inside the first Brillouin zone. The group velocity of a wave packet is given by the gradient of the dispersion:

$\mathbf{v}_g = \frac{1}{\hbar}\nabla_{\mathbf{k}} E(\mathbf{k})$

At zone boundaries, $\nabla_{\mathbf{k}} E$ often goes to zero, meaning the group velocity vanishes and you get standing waves.

Bloch wave functions

Bloch's theorem is the foundation for electrons in periodic potentials. It states that eigenstates of a Hamiltonian with periodic potential take the form:

$\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} \, u_{n\mathbf{k}}(\mathbf{r})$

where $u_{n\mathbf{k}}(\mathbf{r})$ has the same periodicity as the lattice and $n$ is the band index. The plane-wave part $e^{i\mathbf{k}\cdot\mathbf{r}}$ carries the crystal momentum $\hbar\mathbf{k}$, while $u_{n\mathbf{k}}$ encodes how the wave function is modulated within each unit cell.

Because $\psi_{n\mathbf{k}}$ and $\psi_{n,\mathbf{k}+\mathbf{G}}$ describe the same physical state, the quantum number $\mathbf{k}$ only needs to range over the first Brillouin zone. This is what gives rise to the band structure picture: discrete bands $E_n(\mathbf{k})$ defined over the first zone.

Fermi surfaces within Brillouin zones

Electron energy bands and Fermi surfaces

Energy bands form because atomic orbitals broaden into continuous bands of allowed energies when atoms are brought together in a periodic array. The Fermi surface is the constant-energy surface in $\mathbf{k}$-space at the Fermi energy $E_F$, separating occupied from unoccupied states at $T = 0$.

• For a free electron gas, the Fermi surface is a perfect sphere.
• In real metals, the periodic potential distorts the Fermi surface, especially near zone boundaries where gaps open. The Fermi surface can develop necks, pockets, or multiple disconnected sheets.
• The topology of the Fermi surface determines many transport properties: electrical conductivity, Hall coefficient, magnetoresistance, and de Haas–van Alphen oscillations.

Alkali metals (Na, K) have nearly spherical Fermi surfaces that fit well inside the first zone. Noble metals (Cu, Ag, Au) have Fermi surfaces that touch the zone boundary along the $\langle 111 \rangle$ directions, creating the characteristic "necks" at the $L$ points of the FCC zone.

Brillouin zone folding

Extended vs reduced zone schemes

There are two standard ways to display band structures:

• Extended zone scheme: Energy bands are plotted across multiple Brillouin zones, showing how the free-electron parabola gets broken up by gaps at each zone boundary. This is useful for seeing how the periodic potential perturbs the free-electron picture.
• Reduced zone scheme: All bands are folded back into the first Brillouin zone using the relation $\mathbf{k} \to \mathbf{k} + \mathbf{G}$. Each folded branch becomes a separate band with index $n$. This is the standard representation because it compactly shows all bands in one zone.

There's also the periodic zone scheme, where the same band structure is repeated in every zone. It's occasionally useful for visualizing Fermi surface topology across zone boundaries.

The reduced zone scheme is what you'll see in almost every band structure diagram. When you see multiple bands stacked at the same $\mathbf{k}$-point, those come from folding higher-zone states back into the first zone.

Brillouin zone boundaries and degeneracies

High symmetry points and lines

High symmetry points are locations in the Brillouin zone that are invariant under multiple symmetry operations of the crystal's point group. Standard labels for common structures:

• $\Gamma$: always the zone center ($\mathbf{k} = 0$)
• $X$: center of a square face (FCC zone) or face center (SC zone)
• $L$: center of a hexagonal face (FCC zone)
• $K$: midpoint of an edge between hexagonal and square faces (FCC zone)

Along high symmetry lines and at high symmetry points, group theory constrains which bands can cross and which must repel. Degeneracies (multiple states at the same energy) are often protected by symmetry at these special locations. Breaking the symmetry (e.g., by spin-orbit coupling or strain) can lift these degeneracies and open gaps.

Van Hove singularities

Van Hove singularities occur at critical points in the band structure where $\nabla_{\mathbf{k}} E(\mathbf{k}) = 0$. At these points, the density of states (DOS) has a singularity: it diverges (in 2D) or shows a kink/discontinuity (in 3D).

Critical points come in several types depending on the curvature of the band:

• Minimum (all curvatures positive): DOS has a step-like onset in 3D
• Saddle point (mixed curvatures): DOS has a logarithmic divergence in 2D, a kink in 3D
• Maximum (all curvatures negative): DOS drops off with a step

Van Hove singularities matter because a large DOS at the Fermi level enhances electron-electron interactions, which can drive instabilities like superconductivity, magnetism, or charge density waves. They also produce sharp features in optical absorption spectra.

Phonon dispersion in Brillouin zones

Acoustic vs optical phonon branches

Phonon dispersion relations are plotted along high symmetry directions in the Brillouin zone, just like electronic bands. The number of branches depends on the number of atoms per unit cell:

• A unit cell with $p$ atoms in 3D has $3p$ phonon branches total.
• 3 acoustic branches (1 longitudinal, 2 transverse): these have $\omega \to 0$ as $\mathbf{k} \to 0$ (the $\Gamma$ point). Near $\Gamma$, the dispersion is linear: $\omega = v_s |\mathbf{k}|$, where $v_s$ is the sound velocity. All atoms in the unit cell move roughly in phase.
• $3p - 3$ optical branches: these have finite frequency at $\Gamma$. Atoms within the unit cell move out of phase with each other. In ionic crystals, optical phonons couple strongly to infrared light (hence the name "optical").

At the zone boundary, acoustic and optical branches can approach each other, and gaps between them depend on the mass ratio and force constants in the crystal.

Experimental probing of Brillouin zones

X-ray diffraction and Brillouin zones

X-ray diffraction maps out the reciprocal lattice directly. Each diffraction peak corresponds to a reciprocal lattice vector $\mathbf{G}$, and the Bragg condition is equivalent to the Laue condition:

$\mathbf{k}' - \mathbf{k} = \mathbf{G}$

where $\mathbf{k}$ and $\mathbf{k}'$ are the incident and scattered wave vectors. The positions of diffraction peaks give you the reciprocal lattice vectors, from which you can reconstruct the Brillouin zone geometry. Peak intensities encode the structure factor and atomic positions within the unit cell.

Neutron scattering and phonon dispersion

Inelastic neutron scattering is the primary tool for measuring phonon dispersion throughout the Brillouin zone. Neutrons are ideal because their wavelengths and energies are well-matched to interatomic spacings and phonon energies (meV range).

By measuring the energy loss $\hbar\omega$ and momentum transfer $\hbar\mathbf{q}$ of scattered neutrons, you directly map out $\omega(\mathbf{q})$ for each phonon branch. Features to look for in the data include:

• Acoustic branch slopes near $\Gamma$ (giving sound velocities)
• Optical-acoustic gaps
• Soft modes: phonon branches whose frequency drops anomalously, often signaling a structural phase transition
• Kohn anomalies: kinks in phonon dispersion caused by electron-phonon coupling at specific $\mathbf{q}$ vectors related to the Fermi surface